1. Field of the Invention
The invention relates to a method for iterative formation of an image of a specimen to be studied in a particle-optical apparatus.
2. Description of the Related Art
A method of this kind is described in an article by E. J. Kirkland: "Improved high resolution image processing of bright field electron micrographs", Ultramicroscopy 15, (1984), pp. 151-172.
The cited article describes a method for forming an image in an electron microscope by formation of a series of images of a specimen to be examined, each image being formed with a different setting of the focal distance of the imaging lens in the vicinity of the setting for optimum focus. The series thus formed is referred to as a defocus series. The resolution of the microscope can thus be improved in principle by way of calculations performed on the defocus series so as to eliminate the effect of imaging errors. In this context resolution is to be understood to mean the information (in the image or specimen wave) which can be interpreted directly and which is no longer influenced by image artefacts caused by the lens faults. In a field emission TEM (Transmission Electron Microscope), the resolution improvement may be as high as a factor two because of the high degree of coherence in the HRTEM image (High-Resolution TEM), leading to a high information limit. In a TEM equipped with a thermionic source the gain in resolution is less but nevertheless high enough to make sense from a technical point of view. Instead of the focus setting, other relevant particle-optical parameters can be varied, for example the tilting of the electron beam through the objective lens.
As is known, the information concerning the microstructure of a specimen to be studied in an electron microscope is contained in the electron beam whereby the specimen is imaged on a detector (for example, a photographic film or a video camera) by lenses provided in the electron microscope. The wave character of the electrons in the beam describes this information by means of a complex electron wave function .phi.(r) defining the phase and the amplitude of the electron wave as a function of the two-dimensional position vector r perpendicular to the beam in the specimen. In HRTEM direct measurement of the complex electron wave function at the level of the specimen plane is not possible; instead the amplitude squared distribution of the electron wave at the level of the detector is measured, which measurement can be carried out for different settings of the imaging parameters, such as the focal distance of the imaging lens.
In the cited article a first assumption is made as regards the electron wave function .phi.(r) immediately behind the specimen. In the context of the invention it is not of essential importance how this assumption is arrived at. The electron wave function assumed need not be a representation of the actual electron wave function as it occurs when the specimen is actually irradiated, but merely constitutes a starting point for further mathematical processing of an iterative nature and for comparison with actual images of the specimen in order to reconstruct said actual electron wave function (being a representation of the microstructure of the specimen).
The method known from the cited article can be briefly summarized in the form of the following steps a to e:
a On the basis of the assumed electron wave function .phi.(r) a calculation is performed for a series of images which would be obtained by means of this electron wave function for a number of settings of the imaging lens around the setting for optimum focus, i.e. the so-called defocus series. Because this calculation is not based on the actual electron function but on the assumed electron wave function, this process is referred to as the estimation of the defocus series. PA1 b The defocus series thus obtained is compared with the series of experimental images (the defocus series). A so-called differential series is obtained by subtracting the estimated series from the real series. PA1 c Because the aim is to minimize the difference between the estimated series and the real series, a minimalization procedure is applied to the differential series, i.e. the so-called "MAximum Likelihood" procedure (MAL), producing a rule concerning the further iterative processing of the differential series. PA1 d On the basis of the differential series, a calculation is made, using the MAL rule, for a correction function which is used as a correction for the initially assumed electron wave function, thus producing a corrected electron wave function. This calculation of the corrected electron wave function is referred to as the "feedback" step. PA1 e The electron wave function thus corrected can itself serve as a starting point for an iterative processing operation in conformity with the above points a to d, said operations being stopped when the differences between the estimated series and the real series become smaller than a desired value. PA1 1) .phi.(G+G') PA1 2) .phi.*(G') PA1 3) p.sub.nom (G+G') PA1 4) p*.sub.nom (G') PA1 5) E.sub.fs (G+G',G') PA1 .psi..sub.i (R)=(FT).sup.-1 {.psi..sub.i (G)), so that in conjunction with the equation (15) it follows that: EQU .psi..sub.i (R)=(FT).sup.-1 {.psi..sub.i (G)}=(FT).sup.-1 {p.sub.i (G).times..phi.(G)} (16) PA1 h.sub.j is one factor of the set of weighting factors associated with the spatial incoherence,
The above steps of the known method, as implemented in conformity with the cited article, have a number of drawbacks, the drawback of step a will be described first.
The above step a will be described in detail hereinafter. This step is known per se from an article by K. Ishizuka: "Contrast transfer of crystal images in TEM", Ultramicroscopy 5 (1980), pp. 55-65.
The calculation of the estimated defocus series is based on the electron wave function .phi.(r) immediately behind the specimen. It is also assumed that the transfer function p(G) of the optical system of the electron microscope is known. (As is known in this technique, the transfer function of an electron microscope describes the coherent effect of image abberations in the imaging by these apparatus. The "coherent" image abberations concern spherical abberation and the focus setting of the microscope. Incoherent imaging errors must also be taken into account, for example (i) the spread in respect of the directions of incidence of the (non-correlated) electron wave fronts incident on the specimen (known as the spatial incoherence), and (ii) the electronic instabilities in the electron microscope, coupled to the chromatic abberation of the objective lens (known as the temporal incoherence)). As is customary, this transfer function is expressed as a function of the spatial frequencies G(G=.theta./.lambda., in which .theta. is the diffraction deflection angle in the specimen and .lambda. is the wavelength of the electron wave; because the deflection can occur in two independent directions, G has a vector character: G) which can occur in the specimen.
The image quality in a HRTEM is determined to a substantial degree by the so-called temporal incoherence which is caused by instabilities in the electron emission and electron acceleration and in the lens currents. The position of the focal point of the imaging lens, i.e. the imaging in the imaging system, is influenced by a number of variables such as the acceleration voltage of the electron beam, the thermal energy of the electrons in the beam, and the drive current for the objective lens, being the most important imaging (magnetic) lens. Each of these quantities inevitably exhibits a spread with respect to their nominal value (for example, due to noise or thermal spreading), which can be expressed as an additional defocusing with respect to the nominal adjusted focus which can be calculated since the effect of each of these variables on the image is known. The spread in said variables thus causes a spread in the focal distance which exhibits a normal distribution around the adjusted focal distance. Each value of the focal distance which deviates from the nominally adjusted value leads to a respective different value of the transfer function p(G), that is to say p.sub.i (G). This phenomenon, being known as temporal incoherence as stated before, is taken into account in the estimation step in that I(R) (in which I, being the intensity in the image, is dependent on the two-dimensional position vector R in the image) is considered to be the weighted mean value of the sub-images I.sub.i (R) associated with each of the individual transfer functions, the weighting factors being the associated values of the focal distribution which is a normal distribution function. This weighted mean value is mathematically expressed as an integral over the focus parameter of the product of the focal distribution function and the intensity distribution of the sub-images; in a numerical elaboration thereof this integral is replaced by a summing operation as follows: ##EQU1## Numerically the focal distribution function is sampled with a limited series of (2M+1) equidistant focal points with a spacing .epsilon., centred around the nominal focus setting as represented by the maximum of the focal distribution function. For each of these points i, g.sub.i represents the weighting factor of the normal focal distribution function associated with the i.sup.th sub-image I.sub.i (R) of the series of sub-images to be summed. For the sake of simplicity of representation, the expression (1) will be used hereinafter for the analytical integral over the focal distribution function represented by g.sub.i as well as for the numerical representation thereof. In the latter case the index i runs from the minimum value -M to the maximum value +M, so that there are 2M+1 terms in the sum of the expression (1). The set of values g.sub.i, where -M.ltoreq.i.ltoreq.+M (defined on the equidistant focal points with a spacing .epsilon.), is known as the "focal kernel".
The effect of the temporal incoherence is described in the frequency domain by means of the so-called Transmission-Cross-Coefficient (TCC) elaborated in the cited article by Ishizaka in analogy with the wave theory in light optics (see M. Born and E. Wolf "Principles of Optics", Pergamon, London, 1975). Hereinafter the line of thought is described leading, for image estimation, to this formalism as described in expression (12).
In order to enable calculation of the sub-image I.sub.i (R) to be ultimately formed, it must be taken into account that I.sub.i (R) is the modulus squared of the wave function .psi..sub.i (R) at the area of the image, so that EQU I.sub.i (R)=.psi..sub.i (R).times..psi..sub.i *(R)=.vertline..psi..sub.i (R).vertline..sup.2 ( 2)
so the modulus squared of a complex function, .psi..sub.i *(R) being the complex conjugate of .psi..sub.i (R). I.sub.i (R) represents the distribution of the probability of the electron being struck at the detector location having the coordinates R, characterized by the wave function .psi..sub.i (R) which is realised by the transfer function p.sub.i for a focus setting f.sub.i =f+i.epsilon., in which f is the nominal adjusted focal distance. As is generally known from imaging theory, the frequency contents I.sub.i (G) of the sub-image I.sub.i (R) are given by the Fourier transform FT of I.sub.i (R): EQU I.sub.i (G)=FT{I.sub.i (R)}=FT{.psi..sub.i (R).times..psi..sub.i *(R)}=FT{.psi..sub.i (R)}.sym.FT{.psi..sub.i *(R)} (3)
(in which ".sym." is the correlation product between two functions F.sub.1 and F.sub.2 : F.sub.1 (x).sym.F.sub.2 (x) per definition equals .intg.F.sub.1 (x+u).F.sub.2 (u).du).
The Fourier transform of a wave function .psi..sub.i (R) at the area of the detector is written as .psi..sub.i (G), so FT {.psi..sub.i (R)}=.psi..sub.i (G), so that the expression (3) becomes: EQU I.sub.i (G)=.psi..sub.i (G).sym..psi..sub.i *(G) (4)
The transformed wave function .psi..sub.i (G) is the result of the multiplication in the frequency domain of the transfer function by the electron wave function immediately behind the specimen .phi.(G) as can be expressed by the following formula: EQU .psi..sub.i (G)=.phi.(G).times.p.sub.i (G) (5)
The expression (5) describes the propagation through the electron optical lens system of the electron wave from the specimen as far as the detector with a focus setting f.sub.i =f+i.epsilon.. Using (4) and (5), the expression (3) becomes: EQU I.sub.i (G)={.phi.(G).times.p.sub.i (G)}.sym.{.phi.*(G).times.p.sub.i *(G)}(6)
or, by application of the definition of the correlation product to expression (6): EQU I.sub.i (G)=.intg..phi.(G+G').times.p.sub.i (G+G').times..phi.*(G).times.p.sub.i *(G').times.d(G) (7)
Application of the Fourier transformation to expression (1) shows that: ##EQU2## so that by combination of (7) and (8): ##EQU3## which, using a different arrangement, can be written as: ##EQU4##
The form .SIGMA..sub.i {g.sub.i .times.p.sub.i (G+G').times.p.sub.i *(G')} in the expression (10) is referred to as the Transmission Cross-Coefficient (TCC). In conformity with the derivation known from said article by Ishizuka, the TCC is assumed to equal: EQU TCC=p.sub.nom (G+G').times.p.sub.nom *(G').times.E.sub.fs (G+G',G')(11)
in which p.sub.nom (=p.sub.i=0) is the transfer function applicable to the nominal setting f of the focal distance (i.e. without taking into account the temporal incoherence), and E.sub.fs is the envelope function for the focal spread which equals exp[-A{(G+G').sup.2 -(G').sup.2 }.sup.2 ], where A=(.pi..DELTA..lambda.).sup.2 /2, in which .DELTA.=C.sub.c {(.DELTA.V/V).sup.2 +(.DELTA.E/E).sup.2 +4(.DELTA.I/I).sup.2 }.sup.1/2. (C.sub.c =the chromatic abberation constant, V=the acceleration voltage of the electron beam, E=the thermal energy of the electrons upon departure from the electron emitter, I=the drive current of the imaging lens). By insertion of TCC, the expression (10) becomes: EQU I(G)=.intg..phi.(G+G').times..phi.*(G').times.d(G').times.TCC(12)
As has already been stated, the image in an electron microscope is formed by forming a series of K images of a specimen to be examined, each time with a different setting of the focal distance of the imaging lens around the setting for optimum focus. With each (n.sup.th) setting of the lens there is associated a separate TCC, so that in the expression (12) (applicable to the n.sup.th image), in general TCC.sub.n occurs with a total of K different TCCs.
The estimation of an image in the described manner will have to be performed by numerical integration. A problem is then encountered which can be explained as follows. Because of the presence of the TCC.sub.n (dependent on the vectors G and G') as an additional weighting factor in the integration, the integral expression (12) is no longer a pure correlation, so that for numerical integration Fast Fourier Transforms (FFTs) cannot be used so as to circumvent the correlations and the weighted correlation integral must be explicitly calculated. This is then realised as follows: for each value of G it is necessary to run through all values of G', i.e. for each value of G and G' the following expressions must be multiplied:
The above multiplications must be executed for all values of G and G'. For an image comprising N=1000.times.1000=10.sup.6 pixels, the vectors G and G' both have 10.sup.6 values so that, ignoring the determination of the values of the above five expressions, the formation of the product from the explicit correlation already necessitates the execution of a total of N.sup.2 =10.sup.6 .times.10.sup.6 =10.sup.12 multiplications for one image from the defocus series. It is to be noted that, in addition to the phenomenon of temporal incoherence, another, comparable effect occurs in the mathematical description; this effect stems from the spatial incoherence of the irradiating electron beam. This causes a spread in the direction of incidence of the electrons which will be referred to hereinafter as source spread. The description of this spatial incoherence takes the form of an envelop function E.sub.so (G+G',G') for the source spread (E.sub.so =source) which must in principle be added as a factor to the TCC in the equation (11) in conformity with the cited article by Ishizuka. The form of this envelope function is dependent on the type of electron source used and on the focusing conditions of the condenser lens system. When this source is a pure point source (which cannot be realised in practice), E.sub.so .ident.1. If this source is a thermionic source (for example, the known LaB.sub.6 source), a function E.sub.so (G+G',G') must be added. When this source is the field effect source (Field Emission Gun or FEG), because of the property of high coherence of this type of source for E.sub.so in factorized form the following approximate description can be given: E.sub.so (G+G',G')=E.sub.so (G+G',0).times.E.sub.so (0,G'). In the expression (11) the term p.sub.nom (G+G') is then combined with E.sub.so (G+G',0) and the term p*.sub.nom (G') with E.sub.so (0,G'). The two products thus formed can be considered to be a modified form of the original function p.sub.nom (G+G') and of the function p*.sub.nom (G'). The new transfer function thus formed also contains a part which represents the amplitude modulation in the frequency domain due to the spatial incoherence, in addition to the phase modulation due to the lens abberations. This does not make an essential difference in respect of the number of calculation operations to be performed.
The feedback step.
As mentioned sub b, c and d in the above section "The background of the state of the art", the estimated defocus series is subtracted from a defocus series actually detected by means of a detector; this is realised by subtracting the intensity values of corresponding pixels of corresponding images from one another. If desired, such subtraction can take place in the frequency domain, i.e. the values of I.sub.n (G) instead of I.sub.n (R) are subtracted from one another (the index n relates to the n.sup.th item from the defocus series). Thus, a series .DELTA.I.sub.n (G) is formed. From this series the correction function .DELTA..phi.(G) must be calculated so as to be added to the .phi.(G) assumed at the beginning of the iteration cycle (the electron wave function immediately behind the specimen in terms of the spatial frequency) in order to obtain a new .phi.(G) as a starting point for a new iteration step. The correction wave .DELTA..phi.(G) is obtained by averaging over the K individual correction waves of the defocus series .DELTA..phi..sub.n (G), so .DELTA..phi.(G)=(1/K).SIGMA..sub.n .DELTA..phi..sub.n (G). The "new" wave is obtained by summing the "old" wave and the correction wave after multiplication by a factor .gamma. in which .gamma. is a feedback parameter, so that the squared deviation between the real and the estimated defocus series is optimally minimized. Thus, the object is to determine a mathematical procedure for deriving the functions .DELTA..phi..sub.n (G) from the functions .DELTA.I.sub.n (G).
The basis for the above mathematical procedure is a known "Maximum Likelihood" procedure (MAL). According to the state of the art (see the cited article by Kirkland, page 167, formula 66) application of this maximum likelihood procedure results in the following rule to be satisfied by the function .DELTA..phi..sub.n (G): EQU .DELTA..phi..sub.n (G)=.intg..DELTA.I.sub.n (G-G').times..phi.(G').times.d(G').times.TCC.sub.n (G',G) (13)
in which the various expressions under the integral sign have the same meaning as in the formulas (11) and (12). Like in the arithmetic procedure for the estimation (i.e. the numerical elaboration of the equation (12)), the expression (13) is not a pure correlation integral representing the correlation between .DELTA.I.sub.n and .phi.(G), but the integrand is weighted by the TCC. For the execution of the explicit weighted correlation again a multiplication (i.e. the factors under the integral sign in the equation (13)) must be performed for each value of G and G'; this again necessitates N.sup.2 (in the numerical example: 10.sup.12) multiplications for each item of the defocus series.